It's spent. Countering a spell (including a creature) doesn't refund the mana.
701.5. Counter
701.5a To counter a spell or ability means to cancel it, removing it from the stack. It doesn’t resolve and none of its effects occur. A countered spell is put into its owner’s graveyard.
701.5b The player who cast a countered spell or activated a countered ability doesn’t get a “refund” of any costs that were paid.
Yes, cast triggers resolve before the spell that was cast to trigger them. Note that there aren't any cards that currently say "succesfully cast." Anything printed that way says cast in the oracle text.
The stage becomes a non-island dark depths. Continuous abilities like Tidal Warrior's are not taken into account by copy effects. The depths however, is a Legendary Land - Island with only the ability to tap for blue mana, but it's still named dark depths. Since the legend rule cares about types and names, the legend rule still takes effect immediately, and your opponent will choose which land to keep.
If your opponent keeps the stage, it will trigger as normal for the depths/stage combo and your opponent will get Marit Lage.
Here's the relevant section from the tournament rules that answers your questions:
2.11 Taking Notes
Players are allowed to take written notes during a match and may refer to those notes while that match is in
progress. At the beginning of a match, each player’s note sheet must be empty and must remain visible throughout
the match. Players do not have to explain or reveal notes to other players. Judges may ask to see a player’s notes
and/or request that the player explain his or her notes.
Players may not refer to other notes, including notes from previous matches, during games.
Between games, players may refer to a brief set of notes made before the match. They are not required to reveal
these notes to their opponents. These notes must be removed from the play area before the beginning of the next
game. Excessive quantities of notes (more than a sheet or two) are not allowed and may be penalized as slow
play
Yes, until the end of your turn you will have two control changing effects telling you that you control the stolen creature. Keiga's is more recent, so if there were a conflict, it would take precedence.
After your turn ends, the word's effect ends, but Keiga's doesn't, so you maintain control of it.
You get the side of the card that's currently face up. You'd get a Persistent Nightmare from a mirror mockery on one.
When the copy deals combat damage, the token will be returned to your hand and cease to exist before you can do anything with it.
There's no functional difference (although the Chronologist specifically only works on end steps that aren't yours). It's worded that way because it's expected that ball lightning's trigger will happen once, while the chronologist's will happen many times.
I'm not sure about the first two, but your opponent is correct about the third. The tournament shortcut you're proposing when you do multiple things at once is "I do this first thing. Once it's resolved, I do this second thing."
701.5. Counter
701.5a To counter a spell or ability means to cancel it, removing it from the stack. It doesn’t resolve and none of its effects occur. A countered spell is put into its owner’s graveyard.
701.5b The player who cast a countered spell or activated a countered ability doesn’t get a “refund” of any costs that were paid.
If your opponent keeps the stage, it will trigger as normal for the depths/stage combo and your opponent will get Marit Lage.
2.11 Taking Notes
Players are allowed to take written notes during a match and may refer to those notes while that match is in
progress. At the beginning of a match, each player’s note sheet must be empty and must remain visible throughout
the match. Players do not have to explain or reveal notes to other players. Judges may ask to see a player’s notes
and/or request that the player explain his or her notes.
Players may not refer to other notes, including notes from previous matches, during games.
Between games, players may refer to a brief set of notes made before the match. They are not required to reveal
these notes to their opponents. These notes must be removed from the play area before the beginning of the next
game. Excessive quantities of notes (more than a sheet or two) are not allowed and may be penalized as slow
play
After your turn ends, the word's effect ends, but Keiga's doesn't, so you maintain control of it.
When the copy deals combat damage, the token will be returned to your hand and cease to exist before you can do anything with it.
A creature that is both a goblin and a warrior has the creature type goblin, so a goblin obelisk will affect it.